相关论文: The Quenching Problem in the Nonlinear Heat Equati…

This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic…

We investigate in this article the long-time behaviour of the solutions to the energy-dependant, spatially-homogeneous, inelastic Boltzmann equation for hard spheres. This model describes a diluted gas composed of hard spheres under…

In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved.…

In this article, we are interested in studying the Cauchy problems for nonlinear damped wave equations and their systems on a weighted graph. Our main purpose is two-fold, namely, under certain conditions for volume growth of a ball and the…

We study the Cauchy problem for a system of semi-linear coupled fractional-diffusion equations with polynomial nonlinearities posed in $% \mathbb{R}_{+}\times \mathbb{R}^{N}$. Under appropriate conditions on the exponents and the orders of…

In this investigation, symmetry properties of the nonlinear heat conductivity equations of general form $u_t = [E(x, u)u_x]_x + H(x, u)$ are studied. The point symmetry analysis of these equations is considered as well as an equivalence…

We consider the Cauchy problem for the nonlinear Schr\"odinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $\mathbb{R}^d$, $d=5,6$, and energy-critical NLS…

In this paper we consider the long time behavior of solutions of the initial value problem for the damped wave equation of the form \begin{eqnarray*} u_{tt}-\rho(x)^{-1}\Delta u+u_t+m^2u=f(u) \end{eqnarray*} with some $\rho(x)$ and $f(u)$…

We study the existence and uniqueness of source-type solutions to the Cauchy problem for the heat equation with fast convection under certain tail control assumptions. We allow the solutions to change sign, but we will in fact show that…

In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms. Our aim is to obtain the threshold, to classify the global existence of solution for small data or the finite time…

We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the…

This paper concerns with the heat equation in the half-space $\mathbb{R}_{+}^{n}$ with nonlinearity and singular potential on the boundary $\partial\mathbb{R}_{+}^{n}$. We develop a well-posedness theory (without using Kato and Hardy…

We study the behaviour of nonnegative solutions to the quasilinear heat equation with a reaction localized in a ball $$ u_t=\Delta u^m+a(x)u^p, $$ for $m>0$, $0<p\le\max\{1,m\}$, $a(x)=\mathds{1}_{B_L}(x)$, $0<L<\infty$ and $N\ge2$. We…

In our recent precious work, we established the finite time blow up result and upper bound of lifespan estimate to the singular Cauchy problem of semilinear Euler-Poisson-Darboux equation in R^n with subcritical power type nonlinearity. By…

In this manuscript, we solve a nonlinear optimization problem in the study of maximizing cooling temperature using inhomogeneous thermoelectric materials.

In this article we will investigate the large time behavior of solutions of a special class of initial/boundary value problems that involve nonlinear damped beam equations. We will show that the solution energies of global pseudo classical…

We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density $\rho(x)$ and a power-like reaction term $\rho(x) u^p$ with $p>1$; this is a…

We consider time-independent solutions of hyperbolic equations such as $\d_{tt}u -\Delta u= f(x,u)$ where $f$ is convex in $u$. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the…

We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we…

This paper studies the Cauchy problem for a one-dimensional nonlinear peridynamic model describing the dynamic response of an infinitely long elastic bar. The issues of local well-posedness and smoothness of the solutions are discussed. The…